Question

If \( \frac{d}{dx} \left( \frac{x^2+1}{(x^2+5)(x^2+9)} \right) = \frac{2x(x^2+1)}{(x^2+5)(x^2+9)} \left[ \frac{1}{f(x)} - \frac{1}{g(x)} - \frac{1}{h(x)} \right] \), then \( 2h(x)-f(x)-g(x) = \)

• A. 12
• B. 16
• C. 18
• D. 20

Hint:

Use logarithmic differentiation for derivatives of products/quotients: If \(y = u/(vw)\), then \(\ln y = \ln u - \ln v - \ln w\). Differentiating gives \(\frac{y'}{y} = \frac{u'}{u} - \frac{v'}{v} - \frac{w'}{w}\).
Compare the derived expression with the given form to identify \(f(x), g(x), h(x)\).
Substitute these into the target expression.

Answer 1

The Correct Option is A

Let \( y = \frac{u(x)}{v(x)w(x)} \), where \(u(x)=x^2+1\), \(v(x)=x^2+5\), \(w(x)=x^2+9\). We can use logarithmic differentiation. \( \ln y = \ln u - \ln v - \ln w \) Differentiate with respect to x: \( \frac{1}{y} \frac{dy}{dx} = \frac{u'}{u} - \frac{v'}{v} - \frac{w'}{w} \) \( \frac{dy}{dx} = y \left( \frac{u'}{u} - \frac{v'}{v} - \frac{w'}{w} \right) \) Here, \(u = x^2+1 \implies u' = 2x\). \(v = x^2+5 \implies v' = 2x\). \(w = x^2+9 \implies w' = 2x\). So, \( \frac{dy}{dx} = \frac{x^2+1}{(x^2+5)(x^2+9)} \left( \frac{2x}{x^2+1} - \frac{2x}{x^2+5} - \frac{2x}{x^2+9} \right) \) \( \frac{dy}{dx} = \frac{x^2+1}{(x^2+5)(x^2+9)} \cdot 2x \left( \frac{1}{x^2+1} - \frac{1}{x^2+5} - \frac{1}{x^2+9} \right) \) This can be rewritten as: \( \frac{dy}{dx} = \frac{2x(x^2+1)}{(x^2+5)(x^2+9)} \left[ \frac{1}{x^2+1} - \frac{1}{x^2+5} - \frac{1}{x^2+9} \right] \) Comparing this with the given form: \( \frac{d}{dx} (\dots) = \frac{2x(x^2+1)}{(x^2+5)(x^2+9)} \left[ \frac{1}{f(x)} - \frac{1}{g(x)} - \frac{1}{h(x)} \right] \) We can identify: \(f(x) = x^2+1\) \(g(x) = x^2+5\) \(h(x) = x^2+9\) (Note: The order of g(x) and h(x) with the minus signs might be interchangeable if their sum is considered, but the direct comparison suggests this mapping.) We need to find \(2h(x) - f(x) - g(x)\). \(2h(x) - f(x) - g(x) = 2(x^2+9) - (x^2+1) - (x^2+5)\) \( = (2x^2+18) - (x^2+1) - (x^2+5) \) \( = 2x^2+18 - x^2 - 1 - x^2 - 5 \) \( = (2x^2 - x^2 - x^2) + (18 - 1 - 5) \) \( = 0x^2 + (17 - 5) = 12 \). The value is 12. This matches option (a). \[ \boxed{12} \]